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Theorem isclat 15294
Description: The predicate "is a complete lattice." (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
isclat.b  |-  B  =  ( Base `  K
)
isclat.u  |-  U  =  ( lub `  K
)
isclat.g  |-  G  =  ( glb `  K
)
Assertion
Ref Expression
isclat  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  ( dom  U  =  ~P B  /\  dom  G  =  ~P B ) ) )

Proof of Theorem isclat
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 fveq2 5706 . . . . . 6  |-  ( l  =  K  ->  ( lub `  l )  =  ( lub `  K
) )
2 isclat.u . . . . . 6  |-  U  =  ( lub `  K
)
31, 2syl6eqr 2493 . . . . 5  |-  ( l  =  K  ->  ( lub `  l )  =  U )
43dmeqd 5057 . . . 4  |-  ( l  =  K  ->  dom  ( lub `  l )  =  dom  U )
5 fveq2 5706 . . . . . 6  |-  ( l  =  K  ->  ( Base `  l )  =  ( Base `  K
) )
6 isclat.b . . . . . 6  |-  B  =  ( Base `  K
)
75, 6syl6eqr 2493 . . . . 5  |-  ( l  =  K  ->  ( Base `  l )  =  B )
87pweqd 3880 . . . 4  |-  ( l  =  K  ->  ~P ( Base `  l )  =  ~P B )
94, 8eqeq12d 2457 . . 3  |-  ( l  =  K  ->  ( dom  ( lub `  l
)  =  ~P ( Base `  l )  <->  dom  U  =  ~P B ) )
10 fveq2 5706 . . . . . 6  |-  ( l  =  K  ->  ( glb `  l )  =  ( glb `  K
) )
11 isclat.g . . . . . 6  |-  G  =  ( glb `  K
)
1210, 11syl6eqr 2493 . . . . 5  |-  ( l  =  K  ->  ( glb `  l )  =  G )
1312dmeqd 5057 . . . 4  |-  ( l  =  K  ->  dom  ( glb `  l )  =  dom  G )
1413, 8eqeq12d 2457 . . 3  |-  ( l  =  K  ->  ( dom  ( glb `  l
)  =  ~P ( Base `  l )  <->  dom  G  =  ~P B ) )
159, 14anbi12d 710 . 2  |-  ( l  =  K  ->  (
( dom  ( lub `  l )  =  ~P ( Base `  l )  /\  dom  ( glb `  l
)  =  ~P ( Base `  l ) )  <-> 
( dom  U  =  ~P B  /\  dom  G  =  ~P B ) ) )
16 df-clat 15293 . 2  |-  CLat  =  { l  e.  Poset  |  ( dom  ( lub `  l )  =  ~P ( Base `  l )  /\  dom  ( glb `  l
)  =  ~P ( Base `  l ) ) }
1715, 16elrab2 3134 1  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  ( dom  U  =  ~P B  /\  dom  G  =  ~P B ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   ~Pcpw 3875   dom cdm 4855   ` cfv 5433   Basecbs 14189   Posetcpo 15125   lubclub 15127   glbcglb 15128   CLatccla 15292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rex 2736  df-rab 2739  df-v 2989  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-dm 4865  df-iota 5396  df-fv 5441  df-clat 15293
This theorem is referenced by:  clatpos  15295  clatlem  15296  clatlubcl2  15298  clatglbcl2  15300  clatl  15301  oduclatb  15329  xrsclat  26156
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